This
page lists the formulae for cos nx for n=2, up to n=10. By studying the
table of coefficients of these formulae, we can infer other formulae.
For instance, , relating the terms in the table, and enabling us to compute further formulae in an easier manner. See also:
The
following table shows the coefficients of the cosine polynomials for
the appropriate multiple angle. The top of the table shows the powers,
which begin with n and decrease by two for each additional term, if
any. The formulae on the right are meant to clarify the table. The number of each term is k.
k
0
1
2
3
4
5
Power
n
n-2
n-4
n-6
n-8
n-10
Formulae
n
0
1
.
.
.
.
.
cos (0x)=1
1
1
.
.
.
.
.
cos(1x)=cos x
2
2
-1
.
.
.
.
3
4
-3
.
.
.
.
4
8
-8
1
.
.
.
5
16
-20
5
.
.
.
6
32
-48
18
-1
.
.
7
64
-112
56
-7
.
.
8
128
-256
160
-32
1
.
9
256
-576
432
-120
9
.
10
512
-1280
1120
-400
50
-1
Observations from the table
What follows are observations, not all of which are proved on this
page. At this stage, they are hypotheses (if we are thinking
scientically about the data in the table) or conjectures (if we are
thinking mathematically). [They are all provable, however] The
patterns in the table are quite interesting, and they aren't that difficult to understand! We note the following:
Beginning with n=2, the first term coefficients are double the previous one. So, writing T to represent a term, and to refer to the term at line n and column k of the table, with k being 0, 1, ... we note that: , that is, the first term coefficient doubles each time. Also, for n>0, the first term is 2n-1.
We can also note that
,
for k>0 and n>1. For
instance, when n=5, and k=1, we find the
term is 2·(-20)-8, which is -48, where -20 is term 1 when n=4,
and 8 is
term 0 when n=3.
With the cosine (opposite that for the
sine), the highest power is always positive, which is why we write the
formula and the table from the highest power down, to avoid beginning
with a minus sign (there is, of course no mathematical reason to do this!). The subsequent terms alternate in sign; for
instance, term 1 is always negative, term 2, always positive, etc.
The powers are either all even or all odd.
The
final term's coefficient is either ±1 (when the power of
the cosine is 0) or ±n (when the power of the cosine is 1).
The number of terms is ceiling [(n+1)/2] (), for instance for n=5, the number of terms is ceiling ( (5+1)/2)=3
Proof
Let us refer to the polynomial representing cos nx as Tn, each term, k as , where k is term 0, 1... The formula relate to n>1.
We can write cos nx, as Tn, also abbreviating cos x as c, as we would anyway, in our notebooks: [4.1] And the previous two terms, representing cos (n-1)x and cos (n-2)x as: [4.2]
Using 4.3 applied to equations 4.2, we have: [4.3] Note that the coefficient for term k-1 n-2, ,has been added to match up the powers.
Adding like terms, we have: [4.4] ■
First term coefficient is 2n-1
From Equation 4.4,
above, we can deduce that the coefficient for the first term, k=0, is
double the previous one. Because cos (1x)=cos x (the first coefficient
is 1, when n=1), by the formula 4.4, each subsequent first term
coefficient is double, the previous one. So the first term coefficient
for cos 2x is 2, and the first term coefficient for cos(3x) is 4, and
so the nth one is 2n-1.
■
Relationship between the terms
From Equation 4.4, the general term coefficient is precisely our formula: ■ The other observations aren't proved on this page. They can be proved using DeMoivre's theorem and the binomial theorem.
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